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====== Model Comparison ======

Here are the single gender models that have been developed for comparison:

^  Model  ^  [[..:2009:Estrous 1 Gender (Redux)]]  ^  [[Monogamy 1 Gender]]  ^  [[Gestation 1 Gender]]  ^
^  Reactions  |  \( A \xrightarrow{\alpha} E \)\\ \( E \xrightarrow{\gamma} A \)\\ \( N + E \xrightarrow{\beta} N + 2 A \)\\ \( 2 N \xrightarrow{\delta} N \)  |  \( S + S \xrightarrow{\phi} P \)\\ \( P \xrightarrow{\chi} 2 S \)\\ \( P \xrightarrow{\beta} S + P \)\\ \( S + S \xrightarrow{\delta} S \)\\ \( S + P \xrightarrow{2 \delta} P \)\\ \( P + S \xrightarrow{2 \delta} 2 S \)\\ \( P + P \xrightarrow{4 \delta} S + P \)  |  \( 2 A \xrightarrow{\mu} A + G \)\\ \( G \xrightarrow{\beta} 2 A \)\\ \( A + A \xrightarrow{\delta} A \)\\ \( A + G \xrightarrow{\delta} G \)\\ \( G + A \xrightarrow{\delta} A \)\\ \( G + G \xrightarrow{\delta} G \)  |
^  QSSA threshold, \(n_\epsilon\)  |  \( \frac{(\alpha+\gamma)^2}{\abs{\alpha (\beta-\delta) - \gamma \delta}} \)  |  \( \frac{\beta  \chi +2 \chi ^2}{\abs{2 \beta  \phi -2 \delta  \chi }} \)  |  \( \frac{2 \beta}{\abs{\delta -\mu}} \)  |
^  Carrying capacity, \(K\)  |  \( \frac{\alpha \beta - (\alpha + \gamma) \delta }{(\beta + \delta) \delta} \)  |  \( \frac{ 4 r^2 \beta +\beta ^3 - (\beta-2 r) \sqrt{\beta  \left(\beta  (2 r+\beta )^2+8 n_\mathrm{Allee} \left(\beta ^2 - 2 r^2\right) \phi \right)} }{4 (\beta - 2 r)^2 \phi } \)  |  \( \frac{\beta  \left(\delta +2 \mu -\sqrt{\delta  (\delta +8 \mu )}\right)}{2 \delta  \mu } \)  |
^  Allee threshold, \(n_\mathrm{Allee}\)  |  \( \frac{\alpha+\gamma}{\beta+\delta} \)  |  \( \frac{\delta +2 \phi -3 \sqrt{\delta  (\delta +4 \phi )} }{2 \phi ^2 - 16 \delta \phi - 4 \delta ^2 } \chi \)  |  \( \frac{\beta  \left(\delta +2 \mu -3 \sqrt{\delta  (\delta +4 \mu )}\right)}{-2 \delta ^2-8 \delta  \mu +\mu ^2} \)  |
^  Maximum intrinsic growth rate, \( r \)  |  \( \frac{\alpha \beta}{\beta+\delta} \)  |  \( \frac{\beta}{4 \phi }  \left(\delta +2 \phi -\sqrt{\delta  (\delta +4 \phi )}\right) \)  |  \( \frac{\beta  \left(\delta +2 \mu -\sqrt{\delta  (\delta +4 \mu )}\right)}{2 \mu } \)  |

===== Birthrate comparison =====

Each model can be written as \(\diff{n}{t} = B(n) n - \delta n^2\) where the specific birthrate \(B(n)\) is

^  Model  ^  Specific Birthrate \(B(n)\)  ^
^  [[..:2009:Estrous 1 Gender (Redux)]]  |  \( r \frac{n}{n_\mathrm{Allee} + n} \)  |
^  [[Monogamy 1 Gender]]  |  \( \frac{1}{8 n r}\left(-2 n_\mathrm{Allee} r^2+ n_\mathrm{Allee} \beta ^2+n \left(4 r^2+\beta ^2\right)-\sqrt{n^2 \left(-4 r^2+\beta ^2\right)^2+ n_\mathrm{Allee}^2 \left(-2 r^2+\beta ^2\right)^2+2 n n_\mathrm{Allee} \left(-8 r^4+2 r^2 \beta ^2+\beta ^4\right)}\right) \)  |
^  [[Gestation 1 Gender]]  |  \( \frac{1}{4 n r} \left( 2 n \left(r^2+\beta ^2\right)+ n_\mathrm{Allee}  \left(r^2-2 \beta ^2\right) \left(-1+\sqrt{\frac{ n_\mathrm{Allee} -\frac{2 n (r-\beta )^2}{r^2-2 \beta ^2}}{ n_\mathrm{Allee} }} \sqrt{\frac{-2 n (r+\beta )^2+ n_\mathrm{Allee}  \left(r^2-2 \beta ^2\right)}{ n_\mathrm{Allee}  \left(r^2-2 \beta ^2\right)}}\right) \right) \)  |

We can clarify the rates by expressing them in terms of

\[
   x = n / n_\mathrm{Allee} 
\]
\[
   b(x) = B(n) / r. 
\]

Then the dynamics become \(\diff{n}{t} = r b(n/n_\mathrm{Allee}) n - \delta n^2 \) where

^  Model  ^  Reduced birthrate \(b(x) = B(n) / r\)  ^
^  [[..:2009:Estrous 1 Gender (Redux)]]  |  \( \frac{x}{1 + x} \)  |
^  [[Monogamy 1 Gender]]  |  \( \frac{(1+x) \beta^2}{8 r^2 x} - \frac{r^2}{8 r^2 x} \left( 2-4 x+\sqrt{4 (1-2 x)^2+\frac{4 \left(-1+x-2 x^2\right) \beta ^2}{r^2}+\frac{(1+x)^2 \beta ^4}{r^4}}\right) \)  |
^  [[Gestation 1 Gender]]  |  \( \frac{1}{2} \left( 1 + \frac{\beta^2}{r^2} \right) + \frac{r^2-\beta^2}{4 x r^2} \left( -1+\sqrt{1-\frac{2 x (r-\beta )^2}{r^2-2 \beta ^2}} \sqrt{1-\frac{2 x (r+\beta )^2}{r^2-2 \beta ^2}} \right) \)  |

We can further simplify the expressions by recognizing that \(r\) is constrained for the cases of monogamy (\(r<\beta/2\)) and gestation (\(r<\beta\)).

Let's define \( y = r / r_\max \) so that \(0 < y < 1\).  Then

^  Model  ^  Maximum growth rate \(r_\max\)  ^  Reduced birthrate \(b(x,y) = B(n) / r\)  ^
^  [[..:2009:Estrous 1 Gender (Redux)]]  |    |  \( \frac{x}{1 + x} \)  |
^  [[Monogamy 1 Gender]]  |  \(\beta / 2 \)  |  \( \frac{1}{4 x y^2}\left(2 - y^2 + 2 x \left(1+y^2\right) - \sqrt{4 (1+x)^2+4 \left(-1+x-2 x^2\right) y^2+(1-2 x)^2 y^4}\right) \)  |
^  [[Gestation 1 Gender]]  |  \(\beta \)  |  \( \frac{1}{4 x y^2}\left(2 -y^2 + 2 x \left(1+y^2\right) - \sqrt{4 (1+x)^2+4 \left(-1+x-2 x^2\right) y^2+(1-2 x)^2 y^4}\right) \)  |

===== Reduced birthrate =====

Interestingly, monogamy and gestation give //exactly// the same forms for the reduced birthrate \(b(x,y)\).  At the limits \(y=0\) and \(y=1\) it becomes

\[
   b(x,0) = \frac{x}{1+x} 
\]

\[
   b(x,1) = \frac{1+4 x-\sqrt{1+8 x}}{4 x} 
\]

A fair (<5% error) approximation for intermediate values \(0<y<1\) is

\[
   b(x,y) \approx (1-y) b(x,0) + y b(x,1) = \frac{x}{1+x} + \left( \frac{1}{1+x} - \frac{\sqrt{1+8 x} - 1}{4x} \right) y. 
\]

That tells me how the birthrate deviates from the simple Allee effect found in [[..:2009:Estrous 1 Gender (Redux)]].  Is it good for anything?